Throughput and Delay Analysis in Video Streamingover Block-Fading Channels

G. Cocco‡, D. Gunduz∗ and C. Ibars†

‡ German Aerospace Center (DLR), Weßling, Germany

∗ Imperial College, London, United Kingdom

† Intel Corporation, Santa Clara, CA, USA

giuseppe.cocco@dlr.de, d.gunduz@imperial.ac.uk, christian.ibars.casas@intel.com

Abstract—We study video streaming over a slow fading wirelesschannel. In a streaming application video packets are required tobe decoded and displayed in the order they are transmitted asthetransmission goes on. This results in per-packet delay constraints,and the resulting channel can be modeled as a physicallydegraded fading broadcast channel with as many virtual usersas the number of packets. In this paper we study two importantquality of user experience (QoE) metrics, namelythroughputand inter-decoding delay. We introduce several transmissionschemes, and compare their throughput and maximum inter-decoding delay performances. We also introduce a genie-aidedscheme, which provides theoretical bounds on the achievableperformance. We observe that adapting the transmission rateat the packet level, i.e., periodically dropping a subset ofthepackets, leads to a good tradeoff between the throughput andthemaximum inter-decoding delay. We also show that an approachbased on initial buffering leads to an asymptotically vanishingpacket loss rate at the expense of a relatively large initialdelay.For this scheme we derive a condition on the buffering time thatleads to throughput maximization.

I. I NTRODUCTION

Video traffic constitutes a large portion of today’s Internetdata flow, and it is foreseen to exceed70% of the total IPtraffic within the next five years [1]. A significant portion ofthe video traffic is generated by streaming applications, suchas YouTube and Netflix. This, together with the increasingutilization of mobile terminals for streaming high-definitionvideo content, poses growing challenges to mobile networkoperators in terms of bandwidth availability and quality ofuser experience (QoE).

Mobile wireless channels are often modelled with blockfading, where the channel gain stays constant during the chan-nel coherence time, and changes independently across channelblocks according to a certain probability distribution [2]. Fromthe extensive literature on fading channels (see, e.g., [3]-[9]),it emerges that a pivotal role for reliable communicationsis played by the delay constraint, which is a critical designparameter in streaming applications.

In [10] and [11] the broadcast strategy proposed in [12] isused to improve the end-to-end quality in multimedia transmis-sion. However, the broadcast strategy requires encoding bits

into multiple superposed messages of increasing rates, andthislevel of fine adaptation is not possible in practical multimediacommunication systems, in which the encoding rate is fixed bya higher layer application1[13]. Moreover, practical networkarchitectures are strictly layered, and the channel encoder istypically oblivious to the video coding scheme used by theapplication layer; and therefore, rate adaptation is usually notpossible at the code level. Video packets received by thechannel encoder are already video-encoded at a fixed rate,which cannot be changed. On the other hand, the channelencoder can choose to drop some of the video packets, andachieve rate adaptation at the packet level at the expense ofinter-decoding delayat the receiver.

In the Moving Picture Experts Group (MPEG) standard,the video encoder output units are called group of pictures(GOP). Each GOP consists of an I- frame and a number ofP- and B-frames [14]. A GOP can be decoded and displayedindependently of the previous and following GOPs. We assumethat a whole GOP (or an integer number of GOPs) forms onevideo packet, and the coding rate is normalized such that thedisplay time of a GOP (or an integer number of GOPs) isequal to the channel coherence time2.

We consider streaming over a Gaussian block fading chan-nel, in which the transmitter has no channel state information(CSIT), which is the case for networks with large round tripdelay (like satellite networks), or wireless broadcast networkswith a large number of users3. Due to the lack of CSIT,the transmitter uses a fixed transmission rate. In order tominimize the probability of packet loss over the channel, thetransmission rate is kept at the minimum value that allowsno freezing in the display process at the receiver providedno packet is lost. This implies that the transmission time of

1Some streaming protocols, such as HTTP Live Streaming, allow rateadaption among only a limited number of available rates.

2With this we implicitly assume a slow varying channel, for example, amobile terminal moving at pedestrian speed.

3In the downlink channel with many receiving terminals, acquisition ofCSIT is not viable, since this requires the transmission of an extensive amountof information which may result in thefeedback implosionproblem [15].

a packet is equal to its display time (assuming that the timeneeded to process the packet at the receiver is negligible),which is assumed to be constant for all packets. In thestreaming scenario, this imposes a different decoding deadlinefor each video packet, i.e., the first packet needs to be receivedby the end of the first channel block, the second packet by theend of the second block, and so on. Modeling the decoder ateach channel block as a distinct virtual receiver, this channelcan be seen as a physically degraded fading broadcast channelwith as many virtual users as the number of channel blocks.

The loss of a data packet implies the loss of the corre-sponding GOP; and hence, an interruption in the playback ofthe video at the end user, which lasts until the next packet isreceived. In [16] the quality degradation due to GOP lossesas perceived by the end user has been assessed by streamingpre-recorded videos while introducing video segment losses ina controlled fashion. The results illustrate that users aremoretolerant to long freezes with respect to choppy playback, thatis, few long freezing events are on average preferred to manyshort freezing events. However, this is no longer true if thetransmission is for a live event, such as a sport event or newsvideo. In this case, the loss of a large chunk of video content,which may lead to loss of important information, is muchworse than choppy playback quality. In this paper we targetthe latter kind of video content, and consider the interdecodingdelay as a performance measure.

The effect of GOP loss in video streaming has been studiedin [17], [18] and [19]. In the video streaming literature, theproblem is usually tackled at the network level, focusing ontheeffect of packet loss rate, delay and jitter [20]. However, theseparameters are usually assumed to be given as fixed values tothe system designer, or studied from a networking perspective,where packet losses are mainly due to buffer overflow, whilejitter is due to the congestion level of the network, link failuresand dynamic routing. The problem of radio resource allocationin wireless multimedia transmission over frequency selectivechannels is studied in [21] and [22].

We study the interaction between the physical layer andthe display process of the received video data. In particular,we study different communication strategies, each of whichadopts a different policy to select the subset of messagesto be transmitted, as well as the amount of resources (interms of transmission time) dedicated to each message, whichhas an impact on the successful decoding probability. Theperformance of these strategies is evaluated based on twofigures of merit: average throughput and maximum inter-decoding delay [23]. The interaction between the displayprocess and the lower layers is of fundamental importancefor streaming services such as Dynamic Adaptive Streamingover HTTP (DASH), that need an estimation of the link qualityin order to provide an adequate QoE to the end users. In itscurrent implementation DASH uses the information about thelink status at each user in order to optimize the QoE that canbe provided with the available resources [24]. However, DASHsystems require a feedback link that instructs the transmitter onthe highgest bit-rate that can be received in the current channel

condition, whereas we assume no information on the currentchannel state at the transmitter, and thus the optimisationofthe transmission strategy at the transmitter has to be doneindependently of the current channel condition.

While there is an extensive literature on the higher layeranalysis of video streaming applications [25], research onthe physical layer aspects of streaming focus mostly on codeconstruction [26], [27], [28]. The diversity-multiplexing trade-off for a streaming system is studied in [29]. The channelmodel we study here is the dual of the streaming transmittermodel studied in [30], [31], where the data packets, ratherthan being available at the transmitter in advance and havinga per-packet delay constraint, arrive gradually over time,andhave a global delay constraint.

We propose four different transmission schemes based ontime-sharing4. More elaborate transmission techniques havebeen previously studied in literature such as in [10]. In [33]the problem of still images transmitting over slow fadingchannel using a FEC-based multiple description encoder overan OFDM modulation was studied. Unlike in such previousworks, we exclusively focus on time-sharing transmissionbecause of its applicability in practical systems, as it leadsto lower complexity decoding schemes with respect to, forexample, successive interference cancellation, which is re-quired in the case of superposition transmission. Moreover,the throughput and delay analysis is not completely understoodeven for this relatively simpler transmission scheme. In partic-ular, we considermemoryless transmission (MT), equal time-sharing (eTS), pre-buffering (PB)andwindowed time-sharing(wTS)schemes. We also consider an informed transmitter (IT)bound on the achievable throughput and delay performances,assuming perfect CSIT. We compare these achievable schemesand the informed transmitter bound in terms of both through-put and maximum inter-decoding delay. Our results providefundamental performance bounds as well as an insight forthe design of practical video streaming systems over wirelessfading channels.

The rest of the paper is organized as follows. In SectionII we present the system model. In Section III we deriveinformed transmitter bounds on throughput and average maxi-mum delay. In Section IV we presents four different transmis-sion schemes and, for each of them, we analyze throughputand delay. Section V contains the numerical results, while theconclusions are drawn in Section VI.

II. SYSTEM MODEL

We consider a video streaming system over a block fadingchannel. The channel is constant for a block ofn channeluses and changes in an independent and identically distributed(i.i.d.) manner from one block to the next. We assume that thefile to be streamed to the receiver consists ofM independentpackets denoted byW1, . . . ,WM , all available at the transmit-ter at the very beginning. The receiver wants to decode thesepackets gradually as the transmitter continues its transmission.

4Part of the present work has been presented in [32].

Receiver

channel

block M

Receiver

channel

block 2

Receiver

channel

block 1.

Tra

nsm

itte

r

Fig. 1. Equivalent channel model for streaming a video file composed ofM packets overM blocks of the fading channel to a single receiver with aper packet delay constraint.

We assume that the packetWt needs to be decoded by theend of channel blockt, t = 1, . . . ,M , otherwise it becomesuseless. The data packets all have the same size; and it isassumed that each packet is generated at rateR bits perchannel use (bpcu), which is fixed by the application layer,i.e., Wt is chosen randomly with uniform distribution fromthe setWt = {1, . . . , 2nR} [34]. The channel in blockt isgiven by

y[t] = h[t]x[t] + z[t],

whereh[t] is the channel state,x[t] is the length-n channelinput vector,z[t] is a vector of i.i.d. zero mean unit-varianceGaussian noise, andy[t] is the length-n channel output vectorat the receiver. Instantaneous channel states are known only atthe receiver, while the transmitter has only statistical channelknowledge, i.e., it knows the probability density function(pdf)of h(t). We have a short-term average power constraint ofP , i.e., E[x[t]x[t]†] ≤ nP for t = 1, . . . ,M , wherex[t]†

represents the Hermitian transpose ofx[t].The channel from the source to the receiver can be seen

as a physically degraded broadcast channel, such that thedecoder at each channel block acts as a virtual receiver tryingto decode the packet corresponding to that channel block. SeeFig. 1 for an illustration of this channel model. We denote theinstantaneous channel capacity over channel blockt by Ct:

Ct , log2(1 + φ[t]P ), (1)

whereφ[t] = |h[t]|2 is a random variable distributed accordingto a zero-mean pdffΦ(φ). We defineC , E{Ct}, E{x} beingthe mean value ofx.

We define the average throughput,T , as the average decodedrate at the end ofM channel blocks:

T ,R

M

M∑

m=1

m · η(m), (2)

whereη(m) is the probability of decoding exactlym messagesout of M .

In addition to the average throughput, we also study theframe delay, which is defined as the maximum numberof consecutive channel blocks in which the correspondingmessage is not decoded, denoted byDmax. When a videopacket over a channel block is not decoded at the receiver,video playback at the receiver’s device stalls, and the usercontinues to see the same video frame until a new GOP issuccessfully received. SinceDmax is also a random variablewhose realization depends on the channel, we consider theaverage maximum delayD

maxas our performance measure.

We have:

Dmax

,

M∑

d=1

d · Pr{Dmax = d} =

M∑

d=1

Pr{Dmax ≥ d}. (3)

In the next section, we first study an informed transmitterbound on the system performance, assuming perfect CSITabout all the future channel realizations.

III. I NFORMED TRANSMITTER BOUND

An upper bound on the achievable average throughputand a lower bound on the average maximum inter-decodingdelay can be obtained by assuming that the transmitter isinformed about the exact channel realization over all theMchannel blocks non-causally. This allows the transmitter tooptimally allocate the available resources among the messages.In particular, knowing the channelsa priori the transmitter canchoose the optimal subsetSopt of messages to be transmittedthat maximizesT and minimizesD

max. Note that power

allocation across channel blocks is not possible due to short-term power constraint. In order to find the set of messagesSopt that minimizes the average maximum delay, we first findthe maximum number of decodable messages for the givenchannel realizations. It follows from the physically degradedbroadcast channel model depicted in Fig. 1 that the totalnumber of messages that can be decoded up to channel blockt, denoted byΨ(t), t = 1, . . . ,M , is bounded as:

Ψ(t) ≤ min

{

t,

⌊

I tot(t)

R

⌋}

, (4)

whereI tot(t) ,∑t

i=1Ci, is the total mutual information (MI)

accumulated up to and including channel blockt, while ⌊x⌋is the largest integer smaller than or equal tox. At eachchannel blockt, we check whether we can decode packetWt

in addition to the packets that have already been decoded. Notethat there is no gain in decoding a packet prior to its decodingdeadline. Letv(t) ∈ {0, 1} denote the decoding event forWt,i.e., v(t) = 1, if Wt is decoded, andv(t) = 0 if not. We haveΨ(t) = v(1) + · · ·+ v(t), and

v(t+ 1) =

{

1 if I tot(t+ 1) ≥ (Ψ(t) + 1)R,

0 otherwise.(5)

This recursion returns theM -length binary vectorV =[v(1) · · · v(M)], which corresponds to a transmission scheme

= 1 1 0 0 1

Fig. 2. I tot(t) plotted againstt, and the corresponding vectorV in case ofthroughput-optimal transmission. The light blue bars represent the amount ofMI accumulated in each of the5 channel blocks considered, while the darkblue rectangles indicate a decoding event and represent theamount of MI thatis used to decode a message.

that maximizes the throughput. AlthoughV represents anoptimal solution in terms ofT , it may be suboptimal in termsof D.

maxFrom the maximum delay perspective it may be

a better choice not to transmit some of the packets even ifenough mutual information could be accumulated by theirdeadlines, and instead to transmit packets that are furtherinthe sequence. This is equivalent to shifting rightwards some ofthe1’s in V so that the number of consecutive0’s in the vectoris minimized. Note that this process leaves the throughputunchanged.

Let us consider the example shown in Fig. 2, where themutual information accumulated by the receiver at the end ofchannel blockt, I tot(t) is plotted against the channel blocknumber. The linesI tot(t) = jR, j = 1, . . . , 4, indicatethe threshold values ofI tot(t) after which a new messagecan be decoded. The vectorV has entries equal to1 incorrespondence to decoding events (shadowed areas) and zeroin correspondence to channel blocks in which the receiver doesnot decode the corresponding message.

With reference to Fig. 2, the iterative process described byEqn. (5) returns the sequenceV = [11001]. This allocationachieves a throughput of3/5 and a maximum delay of2.However, a better choice for the transmitter is to transmitmessageW3 instead ofW2, as shown in Fig. 3. This gives thenew allocationV′ = [10101], which has the same throughputasV but a maximum delay ofDmax = 1 instead of2.

In order to minimize the maximum delay, the transmittercan choose to drop a message even if it could be decoded withhigh probability. In other words, the resources are allocated toa message with a higher index, which, if decoded, would leadto a lower maximum delay. Note that the maximum delay isoptimized without decreasing the average throughput. Nextwe provide the necessary definitions and results to introducethe algorithmMin_Del_Max_Rate, which optimizes bothT andD

max.

= 1 0 1 0 1

Fig. 3. I tot(t) plotted againstt, and the corresponding vectorV in case ofthroughput- and delay-optimal transmission. The light blue bars represent theamount of MI accumulated in each of the5 channel blocks considered, whilethe dark blue rectangles indicate a decoding event and represent the amountof MI that is used to decode a message.

Definition 3.1: Let Vlb,D denote the binary string of lengthM with maximum number of consecutive zeros equal toD, which has the smallest number of1’s and the smallestdecimal representation.

If M > D, Vlb,D can be constructed by taking a sequenceof M zeros and starting from the(D+ 1)-th most significantbit (i.e., the leftmost one), substituting a0 with a 1, everyDbits. If M = D, Vlb,D is the all-zero string of lengthM .

Let us clarify the definition considering an example withM = 5. To each value ofD in the set {0, 1, 2, 3, 4, 5}corresponds a different vectorVlb,D: Vlb,0 = [11111], Vlb,1 = [01010], Vlb,2 = [00100], Vlb,3 = [00010],Vlb,4 = [00001] andVlb,5 = [00000].

Definition 3.2: We define Ψ(t) =∑t

n=1v(n) and

Ψlb,D(t) =∑t

n=1vlb,D(n), where v(n) and vlb,D(n) are

the n-th bits, starting from the most significant ones, ofV

(tentative allocation vector returned by recursion (5)) andVlb,D (see Definition 1), respectively. In other words,Ψ(t)andΨlb,D(t) are the cumulative sum, from left, of the vectorsV andVlb,D, respectively, up to thet-th coordinate.

With reference to the example in Fig. 2, we haveΨ(1), . . . ,Ψ(5) = 1, 2, 2, 2, 3. For D = 2, we haveVlb,2 = [00100], andΨlb,2(1), . . . ,Ψlb,2(5) = 0, 0, 1, 1, 1.

Theorem 1Given the allocation vectorV returned byrecursion (5), a maximum delay less than or equal toD∗

is achievable if the following holds:Ψ(t) ≥ Ψlb,D∗(t),∀t ∈ {1, . . . ,M}.

Proof We recall thatΨlb,D(t) is the total number of1’s among the leftmostt bits of the sequenceVlb,D (seeDefinition 1), whileΨ(t) is the total number of1’s amongthe leftmost t bits of the sequenceV. Ψ(t) ≥ Ψlb,D(t),∀t ∈ {1, . . . ,M}, implies thatV has at least as many1’s

as Vlb,D among the leftmostt positions,∀t ∈ {1, . . . ,M},which, in turn, implies thatV achieves a maximum delaythat is no greater thanD∗, which concludes the proof.

In order to find the minimum possible maximum delaystarting from a given sequenceV, one can start with a delayD∗ = 0 and check if the condition of Theorem1 is satisfied.If not, the maximum delay is increased by1, and so on.

Algorithm 1 Min_Del_Max_Rate(V)

M = length(V)if V == [0, . . . , 0] then// if no packet can be decoded return the all zero sequence

Sopt = [0, . . . , 0]return Sopt

end ifD, k = 0while found== 0 do

found= 1Vlb,D = [0, . . . , 0] // vector ofM zeros

for i = 1 to⌊

MD+1

⌋

doVlb,D[i(D + 1)] = 1

// assign 1 to thei(D + 1)-th componentend forcumsumd = 0cumsumlb = 0for j = 1 to M do

cumsumd = cumsumd+V[j] // calculateΨ(j)cumsumlb = cumsumlb+Vlb[j]

// calculateΨlb,D(j)if cumsumd < cumsumlb then

// if cumulative sum is lower, start again increasing delayfound= 0exit for

end ifend forif found== 1 then

DmaxIT = D

exit whileend ifD = D + 1

end whileSopt = Vlb,Dmax

IT

excess0 = sum(V)− sum(Vlb,D)while k < excess0 do// assign 1 to the rightmost excess0 zeros ofVlb,Dmax

IT

if Sopt[M − k] == 0 thenSopt[M − k] = 1k = k + 1

end ifend whilereturn Sopt

Using Theorem1, the Min_Del_Max_Rate algorithm(Algorithm 1) has been obtained. The algorithm takes as inputthe vectorV, which is obtained using the recursion in Eqn. (5).First the algorithm calculates the minimum achievable maxi-mum delayDmax

IT (see Theorem1 and the following note) andderives the vectorVlb,Dmax

IT. Then it calculates the difference in

the number of ones betweenV andVlb,DmaxIT

(excess_0 inthe algorithm). By definition ofDmax

IT , excess_0 is greater

than or equal to zero. UsingVlb,DmaxIT

as an initializationallocation vector, the vectorSopt is then constructed by simplysubstituting the rightmostexcess_0 zeros with ones. Theoutput of the algorithm is the set of messagesSopt (containinga 1 or a 0 in positiont if messageWt is to be transmitted, ornot) that constitutes the optimal transmission choice in termsof both throughput and maximum delay. It can be easily shownthat Algorithm 1 has a complexity which is quadratic inM .

In order to clarify the procedure just described, let usconsider again the example in Fig. 2. The recursion in Eqn.(5) returns the vectorV = [11001], which corresponds toΨ =[12223]. The algorithm starts with a tentative delayDmax

IT = 0,and generates the corresponding sequenceVlb,0 = [11111],with Ψlb,0 = [12345]. Since the condition of Theorem1 isnot satisfied (Ψ(3) < Ψlb,0(3)), a minimum maximum delayDmax

IT = 0 cannot be achieved, and the tentative delay isincreased by1, i.e., Dmax

IT = 1. The corresponding sequencesVlb,1 = [01010] andΨlb,1 = [01122] are then calculated. Thecumulative functionΨlb,1 satisfies the condition of Theorem1,which implies that the minimum achievable maximum delayis Dmax

IT = 1. At this point the algorithm calculates the optimalallocation vector. First, the difference in the number of onesbetween vectorVlb,1 and vectorV (excess_0) is computed,which in the example is equal toexcess_0=1. Finally, therightmostexcess_0 zeros inVlb,1 are set to1, which leadsto the allocation sequenceSopt = [01011].

IV. T RANSMISSION SCHEMES

In this section we introduce four different transmissionschemes based on time-sharing. Each channel block is dividedamong the messages for which the deadline has not yetexpired. Thus, while the first channel block is divided amongall the messagesW1, . . . ,WM , the second channel block isdivided among messagesW2, . . . ,WM , as the deadline ofmessageW1 expires at the end of the first block. In generalthe encoder divides channel blockt into M − t + 1 portionsαtt, . . . , αMt, such thatαmt ≥ 0 and

∑M

m=t αmt = 1.In channel blockt, αmtn channel uses are allocated forthe transmission of messageWm. We assume that Gaussiancodebooks are used in each portion for each message, andthe corresponding codelengths are sufficient to achieve theinstantaneous capacity. Then the total amount of receivedmutual information relative to messageWm is:

I totm ,

m∑

t=1

αmtCt. (6)

The proposed schemes differ in the way the channel uses areallocated among the messages for which the deadline has notyet expired. Different time allocations lead to different averagethroughput and average maximum delay performances.

A. Memoryless Transmission (MT)

In memoryless transmission (MT)each message is transmit-ted only within the channel block just before its expiration,that is, messageWt is transmitted over channel blockt.Equivalently we haveαmt = 1, if t = m, and αmt = 0,

otherwise. In MT messageWt can be decoded if and onlyif Ct ≥ R. Due to the i.i.d. nature of the channel state overblocks, the successful decoding probabilityp , Pr{Ct ≥ R}is constant over messages. The probability that exactlymmessages are decoded is given by:

η(m) ,

(

M

m

)

pm(1− p)M−m. (7)

The average number of decoded messages for the MT schemeis TMT = Mp.

Next we derive the exact expression for the averagemaximum delay for MT, denoted byD

maxMT . The term

Pr{Dmax ≥ d} in the summation in Eqn. (3) is theprobability that a sequence ofM Bernoulli random variableswith parameterp contains at leastd consecutive zeros.This probability can be evaluated by modeling the numberof consecutive zeros as a Markov chain, and finding theprobability of reaching the final absorbing state ofdconsecutive zeros. This probability is given in the followingtheorem:

Theorem 2:Let x1, · · · , xM be a sequence of i.i.d. Bernoullirandom variables with parameterp = E[xi]. The probabilityof having at leastd consecutive zeros in the sequence is givenby:

Pr{Dmax ≥ d} =

k∑

i=0

si∑

ri=1

ad,ri

(

M + ri − 1

ri − 1

)(

1

ϕdi

)M

, (8)

wherek ∈ {0, . . . ,M}, k ≤ d + 1 is the number of distinctzeros of the polynomial(1 − z)qd(z) where:

qd(z) = 1− pd∑

j=1

zj(1 − p)j−1, (9)

ϕdi, i ∈ {0, . . . , k}, are the zeros of(1 − z)qd(z) withmultiplicity si, ad,ri, ri ∈ {1, . . . , si}, are constants derivedfrom the partial fraction expansion of

(zp)d

(1 − z)qd(z). (10)

Proof: See Appendix.

Finally, by plugging (8) into (3) we find:

DmaxMT =

M∑

d=1

[

k∑

i=0

si∑

ri=1

ad,ri

(

M + ri − 1

ri − 1

)(

1

ϕdi

)M]

. (11)

B. Equal Time-Sharing (eTS) Transmission

In the equal time-sharing (eTS) transmission scheme eachchannel block is equally divided among all the messageswhose deadline has not expired yet, that is, form = 1, . . . ,M ,we haveαmt =

1

M−t+1for t = 1, . . . ,m, andαmt = 0, for

t = m+ 1, . . . ,M .In eTS, messages whose deadlines are later in time are

allocated more resources; and hence, are more likely to be

decoded. We haveI toti < I tot

j for 1 ≤ i < j ≤ M . Hence, theprobability of decoding exactlym messages is:

η(m) , Pr{I totm ≥ R ≥ I tot

m−1}, (12)

for m = 0, 1, . . . ,M , where we defineI tot0 = 0 and I tot

M+1=

∞. Since the decoded messages in eTS are always the lastones, we can express the average maximum delay of eTS,D

maxeTS, as a function of its average throughputT eTS as follows:

DmaxeTS ,

M∑

m=0

(M −m) · η(m)

=

M∑

m=0

M · η(m)−

M∑

m=0

m · η(m)

= M

(

1−T eTS

R

)

. (13)

The numerical analysis of eTS, together with other schemesis presented in Section V.

C. Pre-Buffering (PB) Transmission

In most practical streaming systems the receiver first accu-mulates GOPs in the playout buffer and then starts displayingthem at a constant frame rate once a sufficient portion of thevideo has been received, in order to compensate for the delayjitter of arriving packets [35]. We consider a slightly differentversion of this type of streaming transmission in which onlythe lastB messages are transmitted while the first packets arenot transmitted at all. The firstM −B+1 channel blocks areused to convey information relative to the lastB packets asexplained in the following. We call this methodpre-buffering(PB) transmission.

The initial buffering phase introduces a start-up delay ofM−B channel blocks. On the other hand, if a sufficiently largebuffering period is chosen, all the transmitted messages can bereceived correctly, achieving an average throughput ofR B

M.

Transmitted messages are encoded with equal time allocationover the firstM −B + 1 blocks. Due to the delay constraint,messageWM−B+1 is transmitted up to channel blockM −B+1. Hence, in blockM−B+2 the lastB−1 messages aretransmitted with equal time allocation. The process continuesup until channel blockM , in which only messageWM istransmitted. Next we indicate withT PB(B) andD

maxPB (B) the

average throughput and the average maximum delay achievedby the scheme using a buffering period ofB channel blocks,respectively. The numberBopt of messages to be transmittedis chosen so that

Bopt = argminB∈{1,...,M}

{

Dmax

(B)}

. (14)

Next we show that theBopt, as defined in Eqn. (14), alsomaximizes the average throughput. The average throughput

when transmitting only the lastB messages is given by:

T PB(B) =R

M

B∑

m=1

Pr {decode at leastm messages}

=R

M

B∑

m=1

Pr{

I totM−m+1 ≥ R

}

, (15)

where the mutual information accumulated by the receiver formessageWm, for m = M − B + 1, M − B + 2, . . . ,M ,is given by:

I totm =

1

B

M−B+1∑

t=1

Ct +m∑

t=M−B+2

Ct

M − t+ 1. (16)

From Eqn. (15) we have:

TPB(B) =R

M

[

B −B∑

m=1

Pr{

I totM−m+1 < R

}

]

=R

M

[

B −B∑

m=1

Pr {Dmax ≥ M −m+ 1}

]

.(17)

The average maximum delay when only the lastB messagesare transmitted is:

DmaxPB (B) = M −B +

∑B

d=1Pr {Dmax ≥ M −B + d} .(18)

From (17) and (18) we find

TPB(B) = R

(

1−D

max(B)

M

)

,

and finally

argminB∈{1,··· ,M}

{

DmaxPB (B)

}

= argmaxB∈{1,··· ,M}

{

T PB(B)}

. (19)

This proves that the average throughput and the maximumdelay can be optimized simultaneously. It is not straight-forward to come up with an analytical expression for theoptimal value ofB in the PB scheme for the general case.In the following theorem we derive the optimal fraction ofmessagesαopt =

Bopt

M, such that almost all of the transmitted

messages can be decoded with probability that approaches1asymptotically asM goes to infinity, if a fractionα′ < αopt

of the messages is transmitted, while a fraction smaller thanαopt of the messages can be decoded ifα′ > αopt.

Theorem 3Average throughput ofαR can be achieved inthe limit of infinite M by transmittingαM + o(M) messagesas long as

α < αopt ,1

R

C+ 1

.

If α > αopt, the achieved average throughput is smaller thanαoptR.

Proof Assume that the last B messages, i.e.,WM−B+1, . . . ,WM , are transmitted, withB = Mα+ o(M),α ≤ 1. MessageWM−B+1, for which the deadline expires

first, is the one that accumulates the least amount of mutualinformation, that is:

IM−B+1 =1

B

M−B+1∑

t=1

Ct. (20)

The probability of decoding all the transmitted messages isthen:

Pr {IM−B+1 ≥ R} = Pr{

1

B

∑M−B+1

t=1Ct ≥ R

}

= Pr{

∑M−B+1

t=1

Ct

M−B+1− C ≥ B

M−B+1R− C

}

= Pr{

SM−B+1 − C ≥ BM−B+1

R− C}

, (21)

whereSM−B+1 ,∑M−B+1

t=1Ct

M−B+1, is the sample mean of

the instantaneous channel capacity over the firstM − B + 1channel blocks. From the law of large numbers it follows that:

limM→∞

Pr{∣

∣

∣SM(1−α− o(M)

M ) − C∣

∣

∣ > δ}

= 0, ∀δ > 0. (22)

Using equations (21) and (22) we find:

limM→∞

Pr {IM−B+1 ≥ R} =

{

1, if limM→∞B

M−B+1R < C

0, if limM→∞B

M−B+1R > C.

(23)

We can write:

limM→∞

B

M −B + 1R = lim

M→∞

Mα+ o(M)

M −Mα+ o(M)R

=α

1− αR. (24)

Finally, using Eqn. (24) in Eqn. (23) we find:

limM→∞

Pr {IM−B+1 ≥ R} =

{

1, if α < αopt

0, if α > αopt.(25)

Eqn. (25) implies that if a fraction of messagesα′ larger thanαopt is transmitted, then the average throughput is less thanαoptR, which concludes the proof.

In Section V, we provide a numerical optimization of the PBscheme, and compare it with the other proposed transmissionstrategies and the upper bound. As we will see from thenumerical results, this buffering approach can improve theaverage throughput significantly as it provides rate adaptationat the packet level by eliminating some of the packets, thusincreasing the correct decoding probability of the remainingpackets.

D. Windowed Time Sharing (wTS)

We have seen in the PB scheme that transmitting only asubset of the messages can improve the system throughput byallowing rate adaptation at the packet level. However, in thePB scheme only the lastB packets are transmitted leadingto a minimum delay ofM − B channel blocks. In the nextscheme, called the windowed time-sharing (wTS) scheme,⌈M/B⌉ messages are transmitted, where⌈x⌉ is the smallestinteger greater than or equal tox; however, unlike in PB,

the transmitted messages are distributed among the whole setof available messages, that is, only one fromB consecutivepackets is transmitted overB consecutive channel blocks. So,for instance, ifB = 3, the first message to be transmitted isW3, which is repeated over channel blocks1, 2 and3, followedby messageW6, which is transmitted in the next three channelblocks, and so on.

The parameterB can be optimized according to two differ-ent criteria, namely to maximize the average throughput or tominimize the delay, which leads to the two variants of the wTSscheme, which we callthroughput-wTS (T-wTS)and delay-wTS (D-wTS), respectively. In wTS a message is decoded withprobabilitypB given below:

pB = Pr {IkB ≥ R} = Pr

min{kB,M}∑

t=kB−W+1

Ck ≥ R

, (26)

for k ∈ {1, . . . ,⌈

MB

⌉

}. A lower bound onDmaxwTS can be found

by substituting⌊

MB

⌋

for M in Eqn. (11),pB for p in equations(9) and (10) and multiplying Eqn. (11) withB. An upperbound can be found in a similar way by using

⌈

MB

⌉

insteadof⌊

MB

⌋

. Similarly, an upper and a lower bound onTwTS

are given by⌈

MB

⌉

· pB and⌊

MB

⌋

· pB, respectively. Analyticaloptimization of parameterB in both the T-wTS and D-wTSschemes is elusive and we resort to the numerical analysispresented in the next section.

V. NUMERICAL RESULTS

In this section we compare the average throughput and theaverage maximum delay of the proposed schemes numerically.The channel model used in the simulations is a Rayleighblock fading channel, in which the channel gainφ[t] inblock numbert, t = 1, . . . ,M (see Eqn. 1) is a unit-meanexponential random variable that changes in an i.i.d. fashionat the beginning of each channel block and stays constantuntil the beginning of the next one. Fig. 4 and Fig. 5 show theaverage throughput and the average maximum delay for theproposed schemes, respectively, forR = 1 and SNR = −5dB. Both variants of the wTS scheme perform close to theinformed transmitter lower bound in terms of the maximumdelay, while the PB scheme is the one with the highest averagethroughput, followed by T-wTS and D-wTS. The eTS schemeshows quite poor performance in terms of both the delayand the throughput. From the plots it emerges that wTS inits two variants T-wTS and D-wTS, can help to reduce theinter-decoding delay while achieving a relatively good averagethroughput in the low SNR regime. The transmitter can choosebetween the two schemes based on its preference betweenhigher throughput and lower inter-decoding delay. While PBprovides the highest throughput among the proposed schemes,its inter-decoding delay is significantly high, due to the initialbuffering time. PB might be a particularly attractive choicefor video streams of long duration, for which the users wouldbe willing to have a larger startup delay to enjoy a higherthroughput for the rest of the video.

10 20 30 40 50 600

0.1

0.2

0.3

0.4

T

M

MTeTSPBT−wTSD−wTSIT

Fig. 4. Average throughputT plotted against the number of messagestransmitted forSNR = −5 dB andR = 1 bpcu.

0 10 20 30 40 50 600

10

20

30

40

50

60

M

Dmax

MTeTSPBT−wTSD−wTSIT

Fig. 5. Average maximum delayDmax

plotted against the number oftransmitted messages forSNR = −5 dB andR = 1 bpcu.

Fig. 6 and Fig. 7 show the average throughput and the av-erage maximum delay, respectively, for the proposed schemesfor R = 1 and SNR = 5 dB. Also for this SNR levelthe two variants of the wTS scheme perform close to theinformed transmitter lower bound in terms of maximum delay.The highest average throughput is achieved by the T-wTSscheme together with the MT scheme, followed by the PB,D-wTS and eTS schemes. From Fig. 6 and Fig. 7 we seethat, when the SNR is high, the MT scheme, together withthe T-wTS scheme, achieves the best performances in termsof both delay and average throughput. This suggests that asimple memoryless approach is sufficient when the channelSNR is sufficiently high, while at low SNR more complexencoding techniques can help to significantly improve theperformance. The D-wTS scheme shows a sudden decreasein the average throughput, which, with reference to Fig. 6,also corresponds to a decrease in the slope of the curve atpoints corresponding toM = 7 and M = 48. This is dueto the optimization of the window sizeB. We recall that inD-wTS the window size represents the number of channelblocks dedicated to a message, and is optimized so as toachieve the minimum average maximum delay. While a largeB leads to a high decoding probability, it implies a small

10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

1.2T

M

MTeTSPBT−wTSD−wTSIT

Fig. 6. Average throughputT plotted against the number of messagestransmitted forSNR = 5 dB andR = 1 bpcu.

5 10 15 20 25 30 35 40 45 50 55 600

5

10

15

20

25

M

Dmax

MTeTSPBT−wTSD−wTSIT

Fig. 7. Average maximum delayDmax

plotted against the number oftransmitted messages forSNR = 5 dB andR = 1 bpcu.

number of transmitted messages, which bounds from belowthe minimum delay byB. As a matter of fact, only⌈M

B⌉

messages are transmitted in the wTS scheme, which impliesthat the maximum delay, in a given realization, is a multipleofB. If, for instance,B = 2 andm = 3 consecutive messagesare lost, the corresponding delay ism · B = 6. Formally,given a window sizeB∗ there is a certain probabilityplB∗

of not decoding a message. For any fixedm ∈ {0, . . . ,M},using Eqn. (8) it can be easily shown that the probability oflosing at leastm consecutive messages increases withM .Thus a valueB∗ which is optimal for a certainM , maynot be the optimal for a larger number of messages, as theprobability that more than one consecutive messages get lostincreases withM . The optimal choice may be to increaseB, so that the probability of losing consecutive messages isdecreased. This is confirmed by Fig. 8, where the optimalwindow size, obtained numerically, is plotted against the totalnumber of messages. An increase inB implies a decrease inthe slope of the average number of decoded messages, sincea smaller fraction of messages is transmitted, as shown inthe plots. The T-wTS scheme, in whichB is optimized so asto achieve the maximum average throughput, shows a goodtradeoff between the average throughput, which, unlike D-

0 10 20 30 40 50 60 70 800

1

2

3

M

Opt

imal

win

dow

siz

e fo

r w

TS

Fig. 8. Optimal window size (B) for the T-wTS scheme plotted versus thetotal number of messages (M ) for SNR = 5 dB.

−20 −15 −10 −5 0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T

SNR [dB]

MTeTSPBT−wTSD−wTSIT

Fig. 9. Average throughputT plotted against theSNR for M = 40 packetsandR = 1 bpcu.

wTS, is almost independent of the number of messages, andthe average maximum delay, performing close to the D-wTSscheme.

In Figures 9 and 10 the average throughput and the aver-age maximum delay, respectively, are plotted against averageSNR. The plots were obtained forM = 40 packets and

−20 −15 −10 −5 0 5 10 15 200

5

10

15

20

25

30

35

40

Dmax

SNR [dB]

MTeTSPBT−wTSD−wTSIT

Fig. 10. Average maximum delayDmax

plotted against theSNR for M =40 packets andR = 1 bpcu.

R = 1 bpcu. As observed in Figures 4 and 6, forM = 40,the PB scheme outperforms all other schemes in terms ofthroughput at low SNR (lower than 2 dB), while T-wTS andMT achieve almost the same performance, and outperform thePB scheme at higherSNRs. From the figures we observe thatthe PB scheme is the most robust one against packet lossesat low SNR, while at higherSNR it is outperformed byall the schemes but the trivial MT. In terms of maximumdelay, PB shows relatively poor performance for most of theconsideredSNR range, which is due to the initial bufferingphase. Note that, if, unlike assumed in this paper, the loss oflarge consecutive chunks of the content were not an issue, andchoppy playback were to be avoided, the PB scheme wouldbe the best among the considered schemes since it guaranteesthat, once the buffering phase is finished, no additional packetis lost, as proven in Theorem 3 for the asymptotic case.

VI. CONCLUSIONS

We have studied the streaming of stored video contentover slow fading channels with per-packet delay constraints.In addition to the classical throughput metric, we have alsoconsidered the inter-decoding delay, i.e., the number of con-secutive video GOPs that cannot be decoded successfully,as a performance measure. We have proposed four differ-ent transmission schemes based on time-sharing. We havecarried out theoretical as well as numerical analysis for theaverage throughput and maximum delay performances. Wehave also derived bounds on both the average throughput andmaximum inter-decoding delay by introducing an informedtransmitter bound, in which the transmitter is assumed to knowthe channel states in advance. We have seen that the wTSscheme can provide a good trade-off between the averagethroughput and the maximum inter-decoding delay by decidingon the proportion of transmitted video packets. In practicethiscorresponds to reducing the coding rate of the video at thepacket level. We have also proved that in the PB scheme almostall transmitted messages can be decoded with a probabilitythat goes to1 asM goes to infinity if only a fraction of themessages smaller than a threshold value, which depends onthe transmission rate and the average channel capacity, aretransmitted.

APPENDIX

Proof of Theorem 1

The probability of having a run of at leastd, d ∈{0, . . . ,M}, consecutive zeros in the sequence is equivalent tofinding the probability of stated afterM steps in the Markovchain depicted in Fig. 11. The stated is an absorbing state,i.e., once the process reaches that state, it remains there withprobability1. Let pt be ad-length probability mass function,wherept(i), i = 0, . . . , d, denotes the probability of being instatei at stept. The vectorpt of state occupancy at stept forthe Markov chain in Fig. 11 can be obtained as:

pt = pt−1H = p0Ht, (27)

Fig. 11. Markov chain for the calculation of the average maximum delay inmemoryless transmission.

wherep0 = [1 0 · · · 0] andH is the(d+1)×(d+1) transitionmatrix of the chain which has the following structure:

H =

1− p p 0 0 · · · 0 01− p 0 p 0 · · · 0 0

......

......

1− p 0 0 0 · · · 0 p0 0 0 0 · · · 0 1

. (28)

The probability of being in stated afterM steps,pM (d), canbe found from Eqn. (27). Sincep0 = [1 0 · · · 0] we have:

pM (d) = HM (1, d+ 1). (29)

In order to evaluateHM (1, d+ 1), we apply theZ-transformto Eqn. (27), taking into account that the recursive formulaisdefined only fort ≥ 1. The Z-transformP(z) of a discretevector functionpt is defined as [36]:

Pz , Z(pt) =+∞∑

t=0

ptzt. (30)

To account for the fact thatt ≥ 1 in Eqn. (27) we can write:

+∞∑

t=1

ptzt =

+∞∑

t=0

ptzt − p0 = Pz − p0, (31)

and+∞∑

t=1

pt−1Hzt = z+∞∑

t=1

pt−1Hzt−1

= z

+∞∑

t=0

ptHzt

= zPzH. (32)

Plugging Eqn. (31) and Eqn. (32) into Eqn. (27) we find:

Pz = p0 (I− zH)−1

, (33)

whereI is the (d+ 1)× (d+ 1) identity matrix.The Z-transformCz of a matrix Ct of functions in the

discrete variablet is defined as:

Cz , Z(Ct) =

+∞∑

t=0

Ctzt. (34)

Note that in Eqn. (34) the termzt is a scalar function ofz andt which is multiplied to each of the elements of matrixCt. Bycomparing Eqn. (33) with Eqn. (27), we see that(I− zH)

−1

is the Z-transform of the matrixHt having functions in thediscrete variablet as elements. We have:

I− zH =

1− z(1− p) −zp 0 0 · · · 0 0−z(1− p) 1 −zp 0 · · · 0 0

......

......

−z(1− p) 0 0 0 · · · 1 −zp0 0 0 0 · · · 0 1− z

.

(35)

Once(I− zH)−1 is known, it is sufficient to inversely trans-

form it and getHt. We find the inverse of matrix (35) for agenericd using Gauss-Jordan elimination. As we only needthe elementHM (1, d + 1), we only report the first row of(I− zH)

−1 in Eqn. (36) at the top of the next page, where

qd(z) , 1− p

d∑

j=1

zj(1 − p)j−1. (37)

The probability of being in stated at stepM is the inverseZ-transform of element(1, d+1) of matrix (I− zH)

−1, i.e.:

pM (d+ 1) = Z−1

{

(zp)d

(1 − z)qd(z)

}

t=M

, (38)

whereZ−1{Pz} is the inverse Z-transform ofPz defined as[36]:

Z−1{Pz} =−1

2πj

∮

γ

Pzz−t−1dz = pt, (39)

γ being a counterclockwise-oriented circle around the originof the complex plane. An easier way to solve Eqn. (38) is todecompose the Z-transform using partial fraction decomposi-tion, i.e., rewritingPz as:

Pz =(zp)d

(1− z)qd(z)=

k∑

i=0

si∑

ri=1

ad,ri

(

1

1− zϕd,i

)ri

, (40)

whereϕd,i, i ∈ {0, . . . , k}, are thek ≤ d + 1 distinct zeroswith degreed+ 1 and multiplicity si of the polynomial(1−z)qd(z), while ad,ri , ri ∈ {1, . . . , si}, are constants derivingfrom the partial fraction expansion ofPz. Once in the form ofEqn. (40),Pz can be inversely transformed using the linearityof the inverse Z-transform and the fact that:

Z−1

{(

1

1− zϕd,i

)ri}

=

(

t+ ri − 1

ri − 1

)(

1

ϕd,i

)t

. (41)

Eqn. (41) follows from the fact that:

Z

{

(

1

ϕ

)t}

,

∞∑

t=0

(

1

ϕ

)t

zt

=

∞∑

t=0

(

z

ϕ

)t

=1

1− z/ϕ, (42)

for |z| < ϕ, and from the fact that the Z-transform of theconvolution of sequences is the product of the Z-transform

of the individual sequences (see [36, Appendix 1] for furtherdetails). Finally, using Eqn. (42) and Eqn. (40) and puttingt = M , we find:

Pr{Dmax ≥ d} =pM (d+ 1)

=

k∑

i=0

si∑

ri=1

ad,ri

(

M + ri − 1

ri − 1

)(

1

ϕdi

)M

.

(43)

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